Additive and multiplicative Gower's Ramsey theorem

Abstract: W. T. Gower generalized Hindman's Finite sum theorem over $X_{k}=\left{ \left(n_{1},n_{2},\ldots,n_{k}\right):n_{1}\neq0\right} $ by showing that for any finite coloring of $X_{k}$ there exists a sequence such that the Gower subspace generated by that sequence is monochromatic. For $k=1,$ this immediately gives the finite sum theorem. In this article we will show that for any finite coloring of $X_{k}$ there exist two sequences $\left{ \mathbf{n_{i}}:i\in I\right} $ and $\left{ \mathbf{m_{i}}:i\in I\right} $ such that the Gower subspace generated by $\left{ \mathbf{n_{i}}:i\in I\right} $ and set of all finite products of $\left{ \mathbf{m_{i}}:i\in I\right} $ are in a single color. This immediately generalize a result of V. Bergelson and N. Hindman which says that for any finite coloring of $\mathbb{N}$, there exist two sequences $\left(x_{n}\right){n}$ and $\left(y{n}\right){n}$ such that the finite sum and product generated by $\left(x{n}\right){n}$ and $\left(y{n}\right)_{n}$ are in a same color.